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\begin{document}
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\title{Optimal Throughput for 802.11 DCF with Multiple Packet Reception}

\author{\IEEEauthorblockN{Mingrui~Zou\IEEEauthorrefmark{1}, Sammy Chan\IEEEauthorrefmark{1}, Hai L.
Vu\IEEEauthorrefmark{2}, Chongbin Xu\IEEEauthorrefmark{3}
and Li Ping\IEEEauthorrefmark{1}}\\
\IEEEauthorblockA{\IEEEauthorrefmark{1}Department of Electronic Engineering, City University of Hong Kong,\\
Tat Chee Avenue, Kowloon Tong, Hong Kong.}
\authorblockA{\authorrefmark{2}Centre for Advanced Internet Architectures, Faculty of ICT, Swinburne Univ. of Technology,\\
P.O. Box 218, VIC 3122, Australia.}
\authorblockA{\authorrefmark{3}Department of Electronic Engineering, Tsinghua University, Beijing, China \\
Email: mrzou2@student.cityu.edu.hk, eeschan@cityu.edu.hk,
h.vu@ieee.org, \\ xcb05@mails.tsinghua.edu.cn, eeliping@cityu.edu.hk}}

\maketitle


\begin{abstract}
In this paper, we propose an analytical model for evaluating the MAC
throughput in an unsaturated IEEE 802.11 wireless local area network
(WLAN) where multiple packets reception (MPR) is possible using
multiuser detection techniques. In particular, a recently proposed successive
interference cancellation (SIC) scheme for MPR is considered where
users can randomly choose the transmission power from a set of discrete power
levels. We derive an explicit expression for throughput of the WLAN
based on such an SIC scheme and validate the accuracy of the model via ns-2
simulation results. We show that the throughput is significantly
improved compared to the conventional 802.11 MAC protocol just by
resolving collisions between two packets with different transmission
power levels. In addition, we provide the optimal power distribution
to maximize the throughput achievable in an SIC-enabled WLAN.
\end{abstract}
%
%\begin{IEEEkeywords}
%WLANs, 802.11, DCF, MUD, MPR.
%\end{IEEEkeywords}


\section{Introduction}~\label{sec:introduction}

Since its inception, IEEE 802.11 wireless local area networks (WLANs)
have attracted a lot of interests. Due to its simplicity and
scalability characteristics, IEEE 802.11 WLANs have become one of
the most deployed worldwide wireless networks which are expected to
play an important role in multimedia home networks and
next-generation wireless communications \cite{QoS}\let\thefootnote\relax\footnotetext
{This work has been performed in the framework of the ICT projects ICT-
217033 WHERE and ICT-248894 WHERE2 which are partly funded by the
European Union, and was also supported by the Research Grants Council of
the Hong Kong Special Administrative Region, China, under Project CityU 111208.}.

In the 802.11 standard, two mechanisms to access the medium are defined.
The fundamental mechanism is the contention-based distributed coordination function (DCF),
while the optional mechanism is the contention-free point coordination function.
The DCF scheme is focused in this paper. DCF is based on the carrier
sense multiple access with collision avoidance (CSMA/CA) multiple access protocol. Collisions
happen when more than one packet are transmitted simultaneously and
result in no useful throughput. However, with the development of multiuser
detection (MUD) techniques \cite{MUD}, it becomes possible to
receive multiple packets which are transmitted simultaneously. For
instance, by using MUD techniques, in CDMA \cite{CDMA} and
multi-antenna systems \cite{multi-antenna}, multiple packet
reception (MPR) can be achieved. There have been various proposed
MUD techniques which include zero-forcing,
minimum-mean-square-error, maximum-likelihood, parallel interference
cancellation and successive interference cancellation (SIC)
 \cite{MUD}.

The development of MUD techniques brings the beneficial MPR capability,
but it also raises two important questions \cite{QZhao2}: how does the
MPR capability affect the MAC protocol performance and how to adjust
the protocol to fully utilize the MPR capability?
There have been many literatures which focused on modeling the
random access networks with MPR capability. In \cite{SGhez1} and
\cite{SGhez2}, a model for general MPR channels was proposed and the
performance for slotted ALOHA with infinite-user assumption was
analyzed. It was extended to finite-user ALOHA system in
\cite{VNaware}. In \cite{QZhao1} and \cite{QZhao2}, the protocols
to exploit the MPR capability were proposed. A Multi-Queue Service
Room (MQSR) protocol was introduced in \cite{QZhao1} which could
yield the optimal utilization of the MPR channel, but with the
drawback of high computational complexity. In \cite{QZhao2},
a much simpler protocol with comparable performance to MQSR was proposed.
In \cite{YJZhang}, the effect of MPR
on the MAC layer collision resolution schemes was studied. It was
shown that the widely used binary exponential backoff scheme did not
yield the close-to-optimal network throughput for both
non-carrier-sensing networks and carrier-sensing networks when
operated in basic access mode of DCF. Furthermore, protocol which considered both
the MAC layer design and the PHY layer signal processing design to
enable MPR capability in distributed random access WLANs was also
proposed.

When SIC is used in high rate applications, its
performance deteriorates when the interfering signals have equal
power. A simple way to address this issue is to use a central
coordinator to allocate different power levels to different users.
However, such a coordinator does not exist in distributed
multi-access systems. Therefore, a practical SIC scheme \cite{CBXu}
is proposed recently to enhance the MPR capability. In this scheme,
power randomization is imposed at
the transmitters of an SIC-based MUD system.  Using the slotted Aloha
as an example, it has been shown that, for the same total power, the
system throughput is significantly improved when this SIC scheme is
used. In this paper, we investigate the throughput improvement of
DCF when SIC is used at the physical layer.

Below we consider an unsaturated IEEE 802.11 network in an infrastructure
topology based on the physical layer utilizing SIC scheme \cite{CBXu}.
To this end, when a node transmits a packet, it randomly
chooses a power level from a set of discrete power levels. Different
from the traditional DCF, collisions between two packets which are
transmitted with different power levels can be resolved. The
throughput performance with this MPR capability is modeled
using a similar approach as in \cite{Sakurai}.
The closed-form expression for
optimal distribution of discrete power levels to maximize system
throughput is also derived.

The remainder of the paper is organized as follows. In Section \ref{sec:SIC},
the SIC scheme with power randomization will be
briefly introduced. In Section \ref{sec:analysis}, the throughput
performance at the MAC layer of IEEE 802.11 WLANs enhanced by such an
SIC scheme is modeled. The accuracy of the model is validated in
Section \ref{sec:validate}. In Section ~\ref{sec:optimization}, the
optimal distribution of discrete power levels to maximize throughput
is derived. Section \ref{sec:conclusion} concludes the paper.


\section{SIC with Power Randomization}\label{sec:SIC}
In this section, the SIC with power randomization (SPR) scheme \cite{CBXu}
is briefly explained.

Consider a 2-user Gaussian multiple-access channel. The received
signal $y$ can be represented by
\begin{equation}\label{eq:y}
y=\sum_{i=1}^2 \sqrt{e_i} x_i +n
\end{equation}
where $x_i$ is the transmitted signal of user $i$ (here $i = 1, 2$)
with power normalized to 1, $e_i$ is power level used to transmit
$x_i$, and $n$ is the complex Gaussian noise with mean 0 and
variance $N_0$. For convenience, we assume that the rate of each
user is equal and denoted as $R$ $(R \geq 1)$. Consider that SIC is deployed in
the receiver. Assume that user 1 is decoded first where the signal
of user 2 is regarded as the noise. For reliable transmission, the
following constraint is required,
\begin{equation}\label{eq:e1}
\log_2(1+\frac{e_1}{e_2+N_0}) \geq R.
\end{equation}
Upon successful decoding, signal of user $1$ is subtracted from $y$.
Then the information for user $2$ can be successfully decoded from
the residual signal after substraction provided that,
\begin{equation}\label{eq:e2}
\log_2(1+\frac{e_2}{N_0}) \geq R.
\end{equation}
Similar discussions can be given to the situation with user 2 being
decoded first.

We now design a transmission strategy based on the constraints in
(\ref{eq:e1}) and (\ref{eq:e2}). Denote by $\mathcal{E}=\{E_i\}$ a
set of positive real values recursively defined below.
\begin{equation}\label{eq:Ei}
    E_i = \begin{cases}
        0 &      \quad i=0 , \\
        (2^R-1)(E_{i-1}+N_0) &   \quad i>0.\\
        \end{cases}
\end{equation}
The set $\mathcal{E}$ has the following two properties.

Property (i): For $i>j$, $E_i$ and $E_j$ satisfy (\ref{eq:e1}),
i.e.,
\begin{equation}
\log_2(1+\frac{E_i}{E_j+N_0}) \geq R.
\end{equation}

Property (ii): For any $j>0$, $E_j$ satisfies (\ref{eq:e2}), i.e.,
\begin{equation}
\log_2(1+\frac{E_j}{N_0}) \geq R.
\end{equation}

Now let the power level of each of users 1 and 2 be randomly drawn
from $\mathcal{E}$. According to Properties (i) and (ii), when two
users are transmitting simultaneously, as long as their chosen power
levels are different, both of their signals can be successfully
decoded.   Note that the situation with $e_i=0$ means user $i$ is not
transmitting and so there is no collision. We will regard this as a
successful case.

Incidentally, it has been proved in \cite{CBXu} that (\ref{eq:Ei})
leads to the optimal power profile based on which the achieved
throughput is not worse than any other profile while less average
power is consumed. In Section \ref{sec:optimization}, based on the
optimal power profile,
we will derive the optimal distribution for those discrete power
levels in $\mathcal{E}$ in order to maximize the system throughput.

\section{Throughput of DCF with SPR} \label{sec:analysis}

There have been many works which focused on modeling the performance
of IEEE 802.11 DCF. In \cite{Bianchi}, a seminal analytical model
was proposed to accurately compute the 802.11 DCF throughput for
saturated networks. Extending the model in \cite{Bianchi}, an
unsaturated model was proposed in \cite{DMalone} which was proved to
have good accuracy. However, the complexity of the model was a
concern since it involves a large state space. A
much simpler model was proposed in \cite{OTickoo2}, but with the
sacrifice of accuracy. In \cite{Sakurai}, a simple model for
unsaturated case was proposed which could obtain comparable accuracy
compared with other existing unsaturated analytical models.

In this section, we extend the model in \cite{Sakurai} to evaluate
the throughput of basic access mode of DCF with the capability of
multiple packet reception enabled by SPR.
Consider an unsaturated
IEEE 802.11 network consisting of $N$ nodes in a infrastructure
mode. The minimum contention window and the retransmission limit
used in DCF are denoted as $W$ and $K$, respectively.

Let $\tau'$ be the attempt rate per slot given that a node has
packet to send and $\tau$ be the unconditional attempt rate per slot for each node.
Also, let $\rho$ be the probability that a node has
packets to send (i.e. its queue is not empty). Hence, we have
\begin{equation}
\tau = \rho \tau'.
\end{equation}

Assuming that packets arrive at a node with rate $\lambda$
[pkts/sec], and that each node has an infinite buffer. Each node can
be modeled as an $G/G/1/\infty$ queue \cite{Sakurai}, with the service time being
determined by the DCF protocol. Let $\overline{S}$ be the average service time,
we have the probability of a nonempty buffer
\begin{equation}
\rho = \lambda \overline{S}.
\end{equation}

Thus the general expression for $\tau$ is given by
\begin{equation}\label{eq:tau}
    \tau = \min(1, \rho) \tau'.
\end{equation}
where the $min()$ function is used to prevent the probability of a
nonempty buffer from exceeding one since $\rho$ might be larger than
one when the network becomes saturated.

First, we need to find the expression for $\tau'$.
Let $\gamma$ be the collision probability experienced by a tagged
node, $R(\gamma)$  and $\overline{W}$ be the average number of attempts and
the mean backoff time (in slots) till a packet transmission is finished
(either successful or not), respectively. Denote by $b_i$ the mean backoff duration (in slots)
at the \emph{i}th attempt, $0 \leq i \leq K$.
With the binary exponential backoff, $b_i$ is given by
\begin{equation}\label{bi}
    b_i = \begin{cases}
            \frac{W}{2} & \quad i=0, \\
            2^i b_0 & \quad 1 \leq i \leq m - 1, \\
            2^m b_0 & \quad m \leq i \leq K, \\
           \end{cases}
\end{equation}
where $m$ determines maximum backoff window size (i.e. $CW_{max} = 2^m W$).
Then it is straightforward to obtain \cite{Kumar}
\begin{align}\label{eq:W}
& R(\gamma) =  1+\gamma+...+\gamma^K, \notag \\
& \overline{W} =  b_0 + \gamma b_1 + ... + \gamma^K b_K.
\end{align}
As in \cite{Kumar}, $\tau'$ can be expressed
as a ratio between $R(\gamma)$ and $\overline{W}$,
\begin{eqnarray}\label{eq:tau'}
    \tau' =  \frac{R(\gamma)}{\overline{W}} = \frac{1+\gamma+..+\gamma^K}{b_0 + \gamma b_1 + .. + \gamma^K b_K}.
\end{eqnarray}

%This can be calculated as a ratio between the mean
%number of attempts per packet and the mean backoff time,
%$\overline{W}$ (in slots), till a packet transmission is finished
%(either successful or not) as follows \cite{Sakurai}
%\begin{eqnarray}\label{eq:tau'}
%    \tau' & = & \frac{(1+\gamma+..+\gamma^K)}{\overline{W}}\\ \nonumber
%          & = & \frac{1+\gamma+..+\gamma^K}{b_0 + \gamma b_1 + .. + \gamma^K b_K}
%\end{eqnarray}
%where $\gamma$ is the collision probability experienced by a tagged
%node, and
%\begin{equation}\label{bi}
%    b_i = \begin{cases}
%            \frac{W}{2} & \quad i=0, \\
%            2^i b_0 & \quad 1 \leq i \leq m - 1, \\
%            2^m b_0 & \quad m \leq i \leq K. \\
%           \end{cases}
%\end{equation}
%where $m$ determines maximum backoff window size (i.e. $CW_{max} = 2^m W$).

When accessing the channel, each node can select a power level from
$\{E_1, E_2, ...\}$ to transmit its packets (i.e. MAC frame). With
SPR, packets can be received successfully if there is no collision,
or collision of two packets that have been transmitted on different
power levels.  Let $p_i$ be the probability that a node chooses to
transmit at power level $E_i$.  The probability $P[E_i\neq E_j]$
that two nodes do not choose the same power levels is given by
\begin{equation}\label{prob-unequal-pwoer}
P[E_i\neq E_j] = 1 - \sum_{i=1}^M p_i^2.
\end{equation}
In the special case that power levels are uniformly distributed,
then
\begin{equation}
P[E_i\neq E_j] = 1 - M(\frac{1}{M})^2
\end{equation}
where $M$ is the number of non-zero powers available.

Let $P_b$ and $P_s$ be the probability that a chosen slot is busy
and that a packet transmission is successful given that there is an
activity in that slot, respectively. We have
\begin{equation}
P_b = 1 - (1-\tau)^N
\end{equation}
\begin{equation}
P_s = \frac{P_1 + P_2 }{P_b}
\end{equation}
where
\begin{equation}\label{eq:P1}
P_1 = {N \choose 1} \tau (1-\tau)^{N-1}
\end{equation}
is the probability that there is only one packet transmitted in a slot,
and
\begin{equation}\label{eq:P2}
P_2 = {N \choose 2} \tau^2 (1-\tau)^{N-2}P[E_i\neq E_j]
\end{equation}
is the probability that two packets are simultaneously transmitted
in a single slot but with different power levels.

To calculate the $\overline{S}$, first we determine the virtual
slot time $T_v$ (in seconds) which is the mean time that elapses for one
decrement of the backoff counter. Considering that, with probability $1 - P_b$,
the slot time is idle; with probability $P_b P_s$, it contains a successful
transmission and with probability $P_b (1 - P_s)$ it contains a collision.
Thus, the virtual slot time $T_v$ is given by \cite{Sakurai}
\begin{equation}\label{eq:Tv}
T_v = (1-P_b)\sigma + P_bP_s (T_s + \sigma) + P_b(1-P_s) (T_c + \sigma)
\end{equation}
where $\sigma$ is an idle slot time,
\begin{equation}
T_s = T_{data} + T_{SIFS} + T_{ACK} + T_{DIFS}
\end{equation}
and
\begin{equation}
T_c = T_{data} + T_{timeout} + T_{DIFS}
\end{equation}
where $T_{timeout} = T_{ACK} + T_{SIFS}$.
$T_{data}$ denotes the transmission time of a data packet, $T_{ACK}$
is the transmission time of an ACK packet, $T_{SIFS}$ and $T_{DIFS}$ represent
the duration of $SIFS$ and $DIFS$, respectively.
The average service time $\overline{S}$ (in seconds) is then
\begin{equation}\label{eq:S}
\overline{S} = \overline{W} T_v.
\end{equation}

Finally $\gamma$ can be calculated as
\begin{equation}\label{eq:gamma}
    \gamma = 1 - (1-\tau)^{N-1} - (N-1)\tau(1-\tau)^{N-2}P[E_i\neq E_j].
\end{equation}

Equations (\ref{eq:tau}), (\ref{eq:tau'}) and (\ref{eq:gamma})  establish
a fixed point from which $\gamma$ can be numerically computed.
%
The channel throughput [bits/second] is
\begin{equation}\label{eq:T}
T = \frac{L P_1 + 2L P_2}{T_v}.
\end{equation}
where $L$ is the packet size in bits.

For easier referencing, the notations of variables used in this paper are summarized as follows,
\begin{center}
\begin{tabular}{lp{0.8\linewidth}}
$N$ \quad & number of nodes.\\
$W$ \quad & minimum contention window size.\\
$K$ \quad &  retransmission limit.\\
$\gamma$ \quad & collision probability.\\
$\tau'$ \quad & attempt rate per slot given that a node has packet to send.\\
$\tau$ \quad & unconditional attempt rate per slot.\\
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{lp{0.8\linewidth}}
$\overline{W}$ \quad & mean backoff time till a packet transmission is finished.\\
$R(\gamma)$ \quad & mean number of attempts till a packet transmission is finished. \\
$m$ \quad & determines maximum backoff window size (i.e. $2^m W$).\\
$\lambda$ \quad & packet arrival rate at a node (pkts/sec).\\
$\overline{S}$ \quad & average service time of a packet (which includes the access delay and transmission time of the packet).\\
$\rho$ \quad & probability that a node has packets to send.\\
$E_i$ \quad & $i$th discrete power level.\\
$p_i$ \quad & probability that a node chooses to transmit at power level $E_i$.\\
$M$ \quad & number of discrete power levels.\\
$P_b$ \quad & probability that a chosen slot is busy.\\
$P_s$ \quad & probability that a packet transmission is successful given that there is an activity in that slot.\\
$P_1$ \quad & probability that there is only one packet transmitted in the slot.\\
$P_2$ \quad & probability that two packets are transmitted in the slot but with different power levels.\\
$\sigma$ \quad & an idle slot time.\\
$T_v$ \quad & virtual slot time.\\
$T_s$ \quad & mean time for a successful transmission.\\
$T_c$ \quad & mean time for an unsuccessful transmission.\\
$T$ \quad & channel throughput.\\
$L$ \quad & packet size.\\
\end{tabular}
\end{center}




\section{Validation}~\label{sec:validate}

\begin{figure*}[t]
\centering
\subfigure{
\begin{minipage}[c]{0.45\textwidth}
\includegraphics[width=0.9\textwidth]{ColProbM3N5.eps}
\includegraphics[width=0.9\textwidth]{ColProbM5N5.eps}
\end{minipage}}
\subfigure{
\begin{minipage}[c]{0.45\textwidth}
\includegraphics[width=0.9\textwidth]{ColProbM3N10.eps}
\includegraphics[width=0.9\textwidth]{ColProbM5N10.eps}
\end{minipage}}
\caption{The collision probability versus normalized total offered
load for N = 5,10 and M = 3,5.}
\label{figure2}
\end{figure*}

\begin{figure*}[t]
\centering
\subfigure{
\begin{minipage}[c]{0.45\textwidth}
\includegraphics[width=0.9\textwidth]{ThrM3N5.eps}
\includegraphics[width=0.9\textwidth]{ThrM5N5.eps}
\end{minipage}}
\subfigure{
\begin{minipage}[c]{0.45\textwidth}
\includegraphics[width=0.9\textwidth]{ThrM3N10.eps}
\includegraphics[width=0.9\textwidth]{ThrM5N10.eps}
\end{minipage}}
\caption{Average throughput versus normalized total offered load for
N = 5,10 and M = 3,5.}
\label{figure1}
\end{figure*}



The model is verified by ns-2 version 2.29. The simulation parameters
are listed in Table \ref{table1}. Average throughput is normalized
by the data transmission rate $R_{data}$. And the normalized offered
load is given by
\begin{equation}
l = \frac{N \lambda L_{pay}}{R_{data}}
\end{equation}
where $L_{pay}$ is the packet payload in bits.

For simplicity, we first study the performance of DCF with SPR under
uniform distribution of discrete power levels, i.e., $P[E_i\neq E_j]
= 1 - M(1/M)^2$. The packet payload is set to be 500 bytes. Fig.
\ref{figure2} and Fig. \ref{figure1} plot the collision probability and
throughput against offered load, respectively, for various $M$ and
$N$.  The accuracy of the proposed performance model is demonstrated
by the agreement between the simulation and analytical results which
are labeled as ``MPR Simulation'' and ``MPR Model'', respectively. Also
included in the figures are the analytical performance results of the original
802.11 DCF protocol, labeled as ``Basic Model''. It can be seen
that, with SPR, the collision probability is decreased while the
average throughput has significant improvement.

Next, we evaluate the performance of DCF with SPR under arbitary
distributions of power levels. Fig. \ref{figure3} plots the
throughput against a set of power level distributions in which the
number of nodes and available power levels is set to be 5 and 3,
respectively. The set contains some combinations of probabilities of
the three power levels. In Fig. \ref{figure3}, the x-axis starts
with the first combination $\{0,0,1.0\}$, which means that the third
power level is always chosen.  The subsequent combinations are
obtained by changing two of the probabilities by a step size 0.1.
For example, the following combinations are $\{0,0.1,0.9\}$,
$\{0,0.2,0.8\}$, $\{0,0.3,0.7\}$ and so on. The arrival rate
$\lambda$ and the packet payload are set to be 250 pkts/sec and 500
bytes, respectively. Again, the accuracy of the proposed model is
demonstrated by the agreement between the simulation and analytical
results.

\begin{table}[t]
\renewcommand{\arraystretch}{1.3}
\small \centering \caption{Simulation parameters for ns-2}
\begin{tabular}[h]{|l|l|}
\hline
CWmin       &  32        \\
\hline
CWmax       &  1024       \\
\hline
Slot Time   &  20 us     \\
\hline
SIFS        &  10 us       \\
\hline
DIFS        &  50 us    \\
\hline
Retry Limit     &  7      \\
\hline
 PHY header  &  192 bits \\
\hline
  MAC header  &  224 bits \\
\hline
Route Header   &   20bytes \\
\hline
 ACK         &  112 bits + PHY header \\
\hline
  Data bit rate     &  11 Mbps \\
\hline
Control bit rate  &  1 Mbps \\
\hline


%   & &  packet payload    &  1020 bytes / 500 bytes \\
%\hline

\end{tabular}
\label{table1}
\end{table}


\section{Optimization}~\label{sec:optimization}

In this section, we aim to find the optimal distribution of the
discrete power levels used at a node that maximizes the overall
throughput. From Section ~\ref{sec:analysis}, the channel throughput
is given in (\ref{eq:T}). Accordingly, we define the following
optimization problem
\begin{equation}\label{eq:opt-prob}
\begin{aligned}
 \text{maximize}  \quad    & L\frac{P_1 + 2P_2}{T_v} \\
 \text{subject to} \quad  & \sum_{i=1}^M p_i = 1 \\
                 & \sum_{i=1}^M p_i E_i = E_{av} \\
                 & 0 \leq p_i \leq 1.
\end{aligned}
\end{equation}
where $E_{av}$ is the average power and $P_1$, $P_2$, $T_v$ are
given in (\ref{eq:P1}), (\ref{eq:P2}) and (\ref{eq:Tv}),
respectively. Substituting these equations into (\ref{eq:T}), the
optimization objective function can be written as
\begin{equation}\label{eq:expaneded-thput}
\begin{aligned}
T = LN \frac{\tau(1-\tau)^{N-1}+
(N-1)\tau^2(1-\tau)^{N-2}(1-\sum\limits_{i=1}^Mp_i^2)}{\sigma +
T_s[1-(1-\tau)^N]}.
\end{aligned}
\end{equation}


In particular, to maximize the channel throughput, we must maximize
(\ref{eq:expaneded-thput}) with respect to the individual
probabilities $p_i$ subject to the constrains that the probabilities
are summed to one and that the total average energy is kept at a
fixed value. This constraint optimization problem can be further
expressed in the equivalent unconstrained form using the
Lagrange multipliers $\alpha$ and $\beta$ as follows.
\begin{equation}\label{eq:Lagrange-thput}
\begin{aligned}
&T(p_1, .., p_M) = \\
&LN \frac{\tau(1-\tau)^{N-1}+ (N-1)\tau^2(1-\tau)^{N-2}(1-\sum\limits_{i=1}^Mp_i^2)}{\sigma + T_s[1-(1-\tau)^N] }\\
& + \alpha (\sum_{i=1}^M p_i-1) + \beta (\sum_{i=1}^M p_i
E_i-E_{av}).
\end{aligned}
\end{equation}



Note that $\tau$ in (\ref{eq:Lagrange-thput}) is a function of $p_i$
through the expression in (\ref{eq:tau}) which is a part of the
fixed point formulation defined in Section \ref{sec:analysis}. To
maximize the function $T(p_1, .., p_M)$ in (\ref{eq:Lagrange-thput})
for each of the individual probabilities $p_i$, let us consider its
partial derivative with respect to the variable $p_i$ at a fixed
$\tau$ value. Given a fixed $\tau$,
\begin{equation}
\frac{\partial T}{\partial p_i} = -2 p_i \frac{LN(N-1)\tau^2(1-\tau)^{N-2}}{\sigma+T_s[1-(1-\tau)^N]} + \alpha + \beta E_i.
\end{equation}


Set $\frac{\partial T}{\partial p_i} = 0$, we obtain
\begin{equation}\label{eq:pi}
p_i = \frac{\alpha + \beta E_i}{2A}
\end{equation}
where
\[A =\frac{LN(N-1)\tau^2(1-\tau)^{N-2}}{\sigma+T_s[1-(1-\tau)^N]}.\]
Substituting (\ref{eq:pi}) into two linear equality constraints in
(\ref{eq:opt-prob}), we obtain
\begin{equation}\label{eq:alpha-beta}
\begin{aligned}
&\frac{M\alpha+E_N\beta}{2A}=1 \\
&\frac{E_N\alpha +E_s\beta }{2A}=E_{av}
\end{aligned}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centering
\includegraphics[width=0.41\textwidth]{GeneMPR.eps}
\caption{Average throughput versus distributions of power levels for N = 5 and M = 3.}
\label{figure3}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where
\[ E_N = \sum_{i=1}^M E_i, ~~~ E_s = \sum_{i=1}^M E_i^2. \]
Solving (\ref{eq:alpha-beta}) for $\alpha$ and $\beta$ we have
\begin{equation}
\begin{aligned}
&\alpha = 2A \left[ \frac{1}{M}-\frac{E_N(E_{av}M-E_N)}{M(ME_s-E_N^2)}\right]  \\
&\beta = 2A \frac{E_{av}M-E_N}{ME_s-E_N^2},
\end{aligned}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{table*}[t]
%\renewcommand{\arraystretch}{1.5}
%\tiny
%\centering
%\caption{Comparison between closed-form expression
%results and optimization solving results for M = 3 and M = 5}
%\begin{tabular}{|c|c|c|c|c|c|}
%\hline
%\multicolumn{3}{|c|}{$M = 3$} & \multicolumn{3}{|c|}{$M = 5$} \\
%\hline
% $E_{av}$ & Closed-form Results  & Optimization Results  &  $E_{av}$ & Closed-form Results  & Optimization Results\\
%\hline
%1.4 & $p_i = \{0.6333,0.3333,0.0333\}$ &  $p_i = \{0.6333,0.3333,0.0333\}$  & 2.4 & $p_i = \{0.32,0.26,0.2,0.14,0.08\}$ &  $p_i = \{0.32,0.26,0.2,0.14,0.08\}$ \\
%%\hline
%    & $T = 0.3644$    & $T = 0.3644$ & &$T = 0.4035$    & $T = 0.4035$ \\
%\hline
%1.6 & $p_i = \{0.5333,0.3333,0.1333\}$ &  $p_i = \{0.5333,0.3333,0.1333\}$ & 2.6 & $p_i = \{0.28,0.24,0.2,0.16,0.12\}$ &  $p_i = \{0.28,0.24,0.2,0.16,0.12\}$ \\
%%\hline
%& $T = 0.3777$    & $T = 0.3777$  &  &$T = 0.4066$    & $T = 0.4066$ \\
%\hline
%1.8 & $p_i = \{0.4333,0.3333,0.2333\}$ &  $p_i = \{0.4333,0.3333,0.2333\}$ & 2.8 & $p_i = \{0.24,0.22,0.2,0.18,0.16\}$ &  $p_i = \{0.24,0.22,0.2,0.18,0.16\}$ \\
%%\hline
%    & $T = 0.3862$    & $T = 0.3861$ &  &$T = 0.4084$    & $T = 0.4084$\\
%\hline
%2 & $p_i = \{0.3333,0.3333,0.3333\}$ &  $p_i = \{0.3333,0.3333,0.3333\}$ &  3 & $p_i = \{0.2,0.2,0.2,0.2,0.2\}$ &  $p_i = \{0.2,0.2,0.2,0.2,0.2\}$\\
%%\hline
%    & $T = 0.389$    & $T = 0.389$ &   &$T = 0.4091$    & $T = 0.4091$\\
%\hline
%2.2 & $p_i = \{0.2333,0.3333,0.4333\}$ &  $p_i = \{0.2333,0.3333,0.4333\}$&  3.2 & $p_i = \{0.16,0.18,0.2,0.22,0.24\}$ &  $p_i = \{0.16,0.18,0.2,0.22,0.24\}$\\
%%\hline
%    & $T = 0.3862$    & $T = 0.3861$  &  &$T = 0.4084$    & $T = 0.4084$\\
%\hline
%2.4 & $p_i = \{0.1333,0.3333,0.5333\}$ &  $p_i = \{0.1333,0.3333,0.5333\}$ &  3.4 & $p_i = \{0.12,0.16,0.2,0.24,0.28\}$ &  $p_i = \{0.12,0.16,0.2,0.24,0.28\}$\\
%%\hline
%    & $T = 0.3777$    & $T = 0.3777$ &  &$T = 0.4066$    & $T = 0.4066$\\
%\hline
%2.6 & $p_i = \{0.0333,0.3333,0.6333\}$ &  $p_i = \{0.0333,0.3333,0.6333\}$  & 3.6 & $p_i = \{0.08,0.14,0.2,0.26,0.32\}$ &  $p_i = \{0.08,0.14,0.2,0.26,0.32\}$\\
%%\hline
%    & $T = 0.3644$    & $T = 0.3644$ & &$T = 0.4035$    & $T = 0.4035$\\
%%\hline
%\hline
%\end{tabular}
%\label{table2}
%\end{table*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
which gives the closed-form expression of $p_i$ as
\begin{equation}\label{eq:pi-detail}
p_i = \frac{E_s-E_{av}E_N+M E_{av}E_i-E_N E_i}{M E_s-E_N^2}.
\end{equation}

Note that the above expression (\ref{eq:pi-detail}) for optimal
$p_i$ is independent of $\tau$ and thus it will also maximize
(\ref{eq:Lagrange-thput}) with an arbitrary $\tau$ value.

In order to validate the accuracy of the obtained results, we
perform a detailed comparison between the derived closed-form
expression results for $p_i$ and that obtained by standard
optimization toolbox \cite{MATLAB} for different number of power
levels ($M$). Results are shown in Table \ref{table2}, where the
following set of parameters is used: $N = 10$, $\lambda = 200$, and
$W_0 = 32$, $m = 5$, $K = 7$, $R = 1$, $N_0 = 1$. It can be seen
from the table that the closed-form derivation gives exactly the
same results as the one by using optimization toolbox.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table*}[t]
\renewcommand{\arraystretch}{1.5}
\tiny
\centering
\caption{Comparison between closed-form expression
results and optimization solving results for M = 3 and M = 5}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{3}{|c|}{$M = 3$} & \multicolumn{3}{|c|}{$M = 5$} \\
\hline
 $E_{av}$ & Closed-form Results  & Optimization Results  &  $E_{av}$ & Closed-form Results  & Optimization Results\\
\hline
1.4 & $p_i = \{0.6333,0.3333,0.0333\}$ &  $p_i = \{0.6333,0.3333,0.0333\}$  & 2.4 & $p_i = \{0.32,0.26,0.2,0.14,0.08\}$ &  $p_i = \{0.32,0.26,0.2,0.14,0.08\}$ \\
%\hline
    & $T = 0.3644$    & $T = 0.3644$ & &$T = 0.4035$    & $T = 0.4035$ \\
\hline
1.6 & $p_i = \{0.5333,0.3333,0.1333\}$ &  $p_i = \{0.5333,0.3333,0.1333\}$ & 2.6 & $p_i = \{0.28,0.24,0.2,0.16,0.12\}$ &  $p_i = \{0.28,0.24,0.2,0.16,0.12\}$ \\
%\hline
& $T = 0.3777$    & $T = 0.3777$  &  &$T = 0.4066$    & $T = 0.4066$ \\
\hline
1.8 & $p_i = \{0.4333,0.3333,0.2333\}$ &  $p_i = \{0.4333,0.3333,0.2333\}$ & 2.8 & $p_i = \{0.24,0.22,0.2,0.18,0.16\}$ &  $p_i = \{0.24,0.22,0.2,0.18,0.16\}$ \\
%\hline
    & $T = 0.3862$    & $T = 0.3861$ &  &$T = 0.4084$    & $T = 0.4084$\\
\hline
2 & $p_i = \{0.3333,0.3333,0.3333\}$ &  $p_i = \{0.3333,0.3333,0.3333\}$ &  3 & $p_i = \{0.2,0.2,0.2,0.2,0.2\}$ &  $p_i = \{0.2,0.2,0.2,0.2,0.2\}$\\
%\hline
    & $T = 0.389$    & $T = 0.389$ &   &$T = 0.4091$    & $T = 0.4091$\\
\hline
2.2 & $p_i = \{0.2333,0.3333,0.4333\}$ &  $p_i = \{0.2333,0.3333,0.4333\}$&  3.2 & $p_i = \{0.16,0.18,0.2,0.22,0.24\}$ &  $p_i = \{0.16,0.18,0.2,0.22,0.24\}$\\
%\hline
    & $T = 0.3862$    & $T = 0.3861$  &  &$T = 0.4084$    & $T = 0.4084$\\
\hline
2.4 & $p_i = \{0.1333,0.3333,0.5333\}$ &  $p_i = \{0.1333,0.3333,0.5333\}$ &  3.4 & $p_i = \{0.12,0.16,0.2,0.24,0.28\}$ &  $p_i = \{0.12,0.16,0.2,0.24,0.28\}$\\
%\hline
    & $T = 0.3777$    & $T = 0.3777$ &  &$T = 0.4066$    & $T = 0.4066$\\
\hline
2.6 & $p_i = \{0.0333,0.3333,0.6333\}$ &  $p_i = \{0.0333,0.3333,0.6333\}$  & 3.6 & $p_i = \{0.08,0.14,0.2,0.26,0.32\}$ &  $p_i = \{0.08,0.14,0.2,0.26,0.32\}$\\
%\hline
    & $T = 0.3644$    & $T = 0.3644$ & &$T = 0.4035$    & $T = 0.4035$\\
%\hline
\hline
\end{tabular}
\label{table2}
\end{table*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusion}~\label{sec:conclusion}

Recently, a successive interference cancellation (SIC) scheme
is proposed to enhance the MPR capability. By adopting this
SIC-based MUD technique at the physical layer, users can
randomly choose the transmission power from a set of discrete power
levels. As a result, collisions between two simultaneously transmitted packets
with different transmission power levels can be resolved.
We proposed a simple but accurate model to evaluate the MAC
performance of an unsaturated IEEE 802.11 WLANS implementing this
MUD technique at the physical layer.
We have validated our analytical model by ns-2 simulation results. It has
been shown that the throughput performance is significantly improved
by resolving collisions between two packets while the collision
probability is dramatically decreased. We have also derived the
closed-form expression for the optimal power distribution which
maximizes the throughput.



%
%In this paper, the throughput performance of an unsaturated IEEE
%802.11 WLANs with MPR capability is modeled. By adopting a recently
%proposed SIC-based MUD technique at the physical layer, users can
%randomly choose the transmission power from a set of discrete power
%levels and collisions between two simultaneously transmitted packets
%with different transmission power levels can be resolved. We have
%validated our analytical model by ns-2 simulation results. It has
%been shown that the throughput performance is significantly improved
%by resolving collisions between two packets while the collision
%probability is dramatically decreased. We have also derived the
%closed-form expression for the optimal power distribution which
%maximizes the throughput.

%\section*{Acknowledgment}
%
%This work was supported by the Research Grants Council of the
%Hong Kong Special Administrative Region, China, under Project CityU 111208.

%\section*{Appendix}


\bibliographystyle{IEEEtranS}
%\bibliographystyle{IEEEtran}
\begin{thebibliography}{1}

\bibitem{QoS}
D. Gu and J. Zhang, ``QoS Enhancement in IEEE802.11 Wireless Local Area Networks,'' \emph{IEEE Commun. Mag.,}
vol. 41, no. 6, pp. 120-124, June 2003.

\bibitem{MUD}
S. Verdu, \emph{Multiuser Detection,} Cambridge Univ. Press, 1998.

\bibitem{CDMA}
P. Rapajic and D. Borah, ``Adaptive MMSE Maximum Likelihood CDMA Multiuser Detection,'' \emph{IEEE J. Selected
Areas Comm.,} vol. 17, no. 12, pp. 2110-2122, December 1999.

\bibitem{multi-antenna}
I. Telatar, ``Capacity of Multi-Antenna Gaussian Channels,'' \emph{European Trans. Telecomm.,} vol. 10, no. 6,
pp. 585-595, November 1999.

\bibitem{QZhao2}
Q. Zhao and L. Tong, ``A Dynamic Queue Protocol for Multiaccess Wireless Networks with Multipacket Reception,''
\emph{IEEE Trans. on Wireless Comm.,} vol. 3, no. 6, pp. 2221-2231, November 2004.
%\bibitem{80211}
%\emph{IEEE Standard for Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specification,} Nov. 1997.

\bibitem{SGhez1}
S. Ghez, S. Verdu, and S. Schwartz, ``Stability Propoerties of Slotted ALOHA with Multipacket Reception Capability,''
\emph{IEEE Trans. Automatic Control,} vol. 33, no. 7, pp. 640-649, July 1988.

\bibitem{SGhez2}
S. Ghez, S. Verdu, and S. Schwartz, ``Optimal Decentralized Control in the Random Access Multipacket Channel,''
\emph{IEEE Trans. Automatic Control,} vol. 34, no. 11, pp. 1153-1163, November 1989.

\bibitem{VNaware}
V. Naware, G. Mergen, and L. Tong, ``Stability and Delay of Finite-User Slotted ALOHA with Multipacket Reception,''
\emph{IEEE Trans. Information Theory,} vol. 51, no. 7, pp. 2636-2656, July 2005.

\bibitem{QZhao1}
Q. Zhao and L. Tong, ``A Multiqueue Service Room MAC Protocol for Wireless Networks with Multipacket Reception,''
\emph{IEEE/ACM Trans. Networking,} vol. 11, no. 1, pp. 125-137, February 2003.

\bibitem{YJZhang}
Y. Zhang, P. Zheng, and S. Liew, ``How Does Multiple-Packet Reception Capability Scale the Performance of Wireless
Local Area Networks?,'' \emph{IEEE Trans. on Mobile Computing,} vol. 8, no. 7, July 2009.

\bibitem{CBXu}
C. Xu , Li Ping, P. Wang, S. Chan, and X. Lin,``Decentralized Power Control for Random Access with Multiple Packets Reception,''
{\it submitted to IEEE Trans. on Wireless Comm.}.

\bibitem{Sakurai}
Q. Zhao, H. K. Tsang, and T. Sakurai, ``A Simple and Approximate Model for Nonsaturated IEEE 802.11 DCF,''
\emph{IEEE Trans. on Mobile Computing,} vol. 8, no. 11, November 2009.


\bibitem{Bianchi}
G. Bianchi, ``Performance Analysis of the IEEE 802.11 Distributed Coordination Function,'' \emph{IEEE J. Selected
Area in Comm.,} vol. 18, no. 3, pp. 535-547, March 2000.



\bibitem{DMalone}
D. Malone, K. Duffy, and D. Leith, ``Modeling the 802.11 Distributed Coordination Function in Non-Saturated Heterogeneous
Conditions,'' \emph{IEEE/ACM Trans. Networking,} vol. 15, no. 1, pp. 159-172, February 2007

%\bibitem{OTickoo1}
%O. Tickoo and B. Sikdar, ``Queueing Analysis and Delay Mitigation in IEEE 802.11 Random Access MAC Based Wireless Networks,''
%\emph{Proceedings of IEEE INFOCOM,} 2004.

\bibitem{OTickoo2}
O. Tickoo and B. Sikdar, ``A Queueing Model for Finite Load IEEE 802.11 Random Access MAC,'' \emph{Proceedings of IEEE ICC,}
2004.

\bibitem{Kumar}
A. Kumar, E. Altman, D. Miorandi, and M. Goyal, ``New Insights from a Fixed Point Analysis of Single Cell IEEE 802.11 WLANs,''
\emph{IEEE/ACM Trans. Networking,} vol. 15, no. 3, pp. 588-601, Mar. 2007.

\bibitem{MATLAB}
\emph{MATLAB 7.6.0 (R2008a).}
\end{thebibliography}




\end{document}
